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RSolveValue

gives the value of expr determined by a symbolic solution to the ordinary difference equation eqn with independent variable n.

RSolveValue[{eqn₁,eqn₂,…},expr,…]

uses a symbolic solution for a list of difference equations.

RSolveValue[eqn,expr,{n₁,n₂,…}]

uses a solution for the partial recurrence equation eqn.

Details and Options

RSolveValue[eqn,a,n] gives a solution for a as a pure function.
The equations can involve objects of the form a[n+λ], where λ is a constant, or in general, objects of the form a[ψ[n]], a[ψ[ψ[n]]], a[ψ[…[ψ[n]]…]], where ψ can have forms such as:

	n+λ	arithmetic difference equation
	μ n	geometric or -difference equation
	μ n+λ	arithmetic-geometric functional difference equation
	μ n^α	geometric-power functional difference equation
		linear fractional functional difference equation

Equations such as a[0]==val can be given to specify end conditions.
If not enough end conditions are specified, RSolveValue will use general solutions in which undetermined constants are introduced.
The specification a∈Vectors[p] or a∈Matrices[{m,p}] can be used to indicate that the dependent variable a is a vector-valued or a matrix-valued variable, respectively. Alternatively, a can be specified as a VectorSymbol or MatrixSymbol. » »
The constants introduced by RSolveValue are indexed by successive integers. The option GeneratedParameters specifies the function to apply to each index. The default is GeneratedParameters->C, which yields constants C[1], C[2], ….
GeneratedParameters->(Module[{C},C]&) guarantees that the constants of integration are unique, even across different invocations of RSolveValue.
For partial recurrence equations, RSolveValue generates arbitrary functions C[n][…].
Solutions given by RSolveValue sometimes include sums that cannot be carried out explicitly by Sum. Dummy variables with local names are used in such sums.
RSolveValue[eqn,a[Infinity],n] gives the limiting value of the solution a at Infinity.
RSolveValue handles both ordinary difference equations and ‐difference equations.
RSolveValue handles difference‐algebraic equations, as well as ordinary difference equations.
RSolveValue can solve linear recurrence equations of any order with constant coefficients. It can also solve many linear equations up to second order with nonconstant coefficients, as well as many nonlinear equations.
RSolveValue[u[t]sys,resp,t] can be used for solving discrete-time models, where sys can be a TransferFunctionModel or a StateSpaceModel and the response function resp can be one of the following: »
"StateResponse" state response of sys to the input

"OutputResponse" output response of sys to the input

Examples

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Basic Examples (4)

Solve a difference equation:

Include a boundary condition:

Get a "pure function" solution for a:

Plot the solution:

Solve a functional equation:

Obtain the value of the solution at a point:

List of values:

Scope (54)

Basic Uses (9)

Compute the general solution of a first-order difference equation:

Obtain a particular solution by adding an initial condition:

Plot the solution of a first-order difference equation:

Make a table of values:

Verify the solution of a difference equation by using a in the second argument:

Obtain the general solution of a higher-order difference equation:

Particular solution:

Solve a system of difference equations:

Plot their solution:

Verify the solution:

Compute the value of the solution at a point:

Compute the limiting value of the solution at Infinity:

Solve a partial difference equation:

Obtain a particular solution:

Plot the resulting solution:

Use different names for the arbitrary constants in the general solution:

Linear Difference Equations (7)

Geometric equation:

First-order equation with variable coefficients:

A third-order constant coefficient equation:

Initial value conditions:

Plot the solution:

Second-order inhomogeneous equation:

Second-order variable coefficient equation in terms of elementary functions:

Euler–Cauchy equation:

In general, special functions are required to express solutions:

Higher-order inhomogeneous equation with constant coefficients:

Nonlinear Difference Equations (5)

Solvable logistic equations:

Riccati equations:

Solutions in terms of trigonometric and hyperbolic functions:

Higher-order equations:

Nonlinear convolution equation:

Systems of Difference Equations (8)

Linear system with constant coefficients:

With boundary conditions:

Plot their solution:

Linear fractional systems:

Diagonal system:

Variable coefficient linear system with a polynomial solution:

Linear constant coefficient difference-algebraic system:

An index-2 system:

Solve a linear system using vector variables:

Alternatively, define as a VectorSymbol:

Solve a linear system using matrix variables:

Alternatively, define as a MatrixSymbol:

Solve an inhomogeneous linear system of ODEs with constant coefficients:

Partial Difference Equations (3)

First-order linear partial difference equation with constant coefficients:

Substitute the function Sin[2k] for the free function C[1]:

Plot the resulting solution:

Constant coefficient linear equation of orders 2, 3, and 4:

Inhomogeneous:

Variable coefficient linear equation:

Q–Difference Equations (6)

First-order constant coefficient -difference equation:

Equivalent way of expressing the same equation:

Initial value:

Second-order equation:

Third-order:

Inhomogeneous:

Using a numeric value for :

Plot solution:

Linear varying coefficient equations:

Nonlinear equations:

Riccati equation:

A linear constant coefficient system of -difference equations:

Functional Difference Equations (4)

Find the general solution for an arithmetic difference equation:

Verify the solution:

Solve an initial value problem for an arithmetic-geometric difference equation:

Plot the solution:

Solve a linear fractional difference equation:

Make a table of values for the solution:

Solve a geometric power difference equation:

Verify the solution: